A systematic analysis of the X-ray spectra (0.1-10 keV) of all blazars observed with EXOSAT is presented. The objects include 16 X-ray-selected BL Lac objects (XBLs), five radio-selected BL Lac objects (RBLs), and five highly polarized quasars (HPQs). Comparing the results of single-power-law fits for the different classes, we find that XBLs have steeper X-ray spectra than HPQs (average photon indices [GAMMA] approximately 2.20(-0.15)+0.17 and 1.62(-0.24)+0.11, respectively). The X-ray luminosities of XBLs and RBLs are similar, while HPQs tend to be somewhat more luminous. For the three groups, a correlation between the average medium-energy luminosity and the average spectral index is found, in the sense that sources of higher luminosity have flatter spectra. For repeatedly observed objects (XBLs), the spectra harden systematically with increasing intensity. For seven (out of 16) XBLs, corresponding to the brightest objects in the sample, the X-ray spectrum is significantly better described by a convex, broken-power-law model than by a single power law. The results are discussed in the framework of current models for blazars.